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of a young and vigorous nation which had yet to outgrow its
morbid addiction to senatorial oratory and Fourth of July
verbal fireworks.
Quaternions has too long a history for the whole story to be
told here. Even Gauss with his anticipation of 1817 was not the
first in the field: Euler preceded him with an isolated result
which is most simply interpreted in terms of quaternions. The
origin of quaternions may go back even farther than this, for
Augustus de Morgan once half-jokingly offered to trace their
history for Hamilton from the ancient Hindus to Queen Vic-
toria. However, we need glance here only at the lion's share in
the invention and consider briefly what inspired Hamilton.
The British school of algebraists, as will be seen in the chapter
on Boole, put common algebra on its own feet during the first
half of the nineteenth century. Anticipating the currently
accepted procedure in developing any branch of mathematics
carefully and rigorously they founded algebra postulationally.
Before this, the various kinds of 'numbers' - fractions, nega-
tives, irrationals - which enter mathematics when it is assumed
that all algebraic equations have roots, had been allowed to
function on precisely the same footing as the common positive
integers which were so staled by custom that all mathemati-
cians believed them to be 'natural' and in some vague sense
completely understood - they are not, even to-day, as will be
seen when the work of Georg Cantor is discussed. This naive
faith in the self-consistency of a system founded on the blind,
formal juggling of mathematical symbols may have been
sublime but it was also slightly idiotic. The climax of this
credulity was reached in the notorious principle of permanence
of form, which stated in effect that a set of rules which yield
consistent results for one kind of numbers - say the positive
integers - will continue to yield consistency when applied to any
other kind - say the imaginaries - even when no interpretation
of the results is evident. It does not seem surprising that this
faith in the integrity of meaningless symbols frequently led to
The British school changed all this, although they were
unable to take the final step andprore that their postulates for